From: greg@math.ucdavis.edu (Greg Kuperberg)
Subject: This week in the mathematics arXiv (11 Sep - 15 Sep)
Date: 20 Sep 2000 07:30:02 -0500
Newsgroups: sci.math.research
Summary: [missing]

[deletia --djr]

This week I'm going to renege on my own mathematical discussion in favor
of an exuberant Featured Review, by Voronov, of the article "Operads
and motives in deformation quantization" [math.QA/9904055], by Kontsevich:

    http://www.ams.org/msnhtml/featured-reviews/2000j53119_rev.html

To summarize the beginning of the review, the original problem that
Kontsevich solved was deformation quantization of Poisson manifolds
[q-alg/9709040]. This is one rigorous interpretation of the old issue
of quantum mechanics known as "canonical quantization", namely how to
replace a classical Hamiltonian with a quantum-mechanical one.  In the
mathematical version, the space on which the Hamiltonian is defined
becomes a Poisson manifold.  A coherent picture of quantizations of all
Hamiltonians requires a deformation quantization, i.e., a deformation of
the commutative algebra of coordinate functions on the Poisson manifold
to a non-commutative algebra, where the commutators are to leading order
given by the Poisson structure.  The first big question is whether a
deformation quantization always exists; Kontsevich proved that it does.
Kontsevich proves a more general result called the "formality theorem".
In this sequel article, Kontsevich applies formality theory, as only he
can, to many other areas of mathematics, from motivic number theory to
perturbative quantum field theory.

Both the article and the review signal the state of scholarly
communication in mathematics in several ways. Kontsevich's article
has already had a long life in the arXiv; indeed, I discussed it in
"This week..." when it was first contributed.  Math Reviews has taken
notice because it has since been published in the traditional sense
[Lett. Math. Phys. 48 (1999), 35-72].  As it happens, it is the
explicit joint policy of Math Reviews and Zentralblatt not to review
arXiv articles unless and until they are traditionally published.
This is why q-alg/9709040, which was Kontsevich's predessor article,
isn't indexed in Math Reviews, although as consolation it is cited in
the text of Voronov's review.  Another unusual policy illustrated by the
review is that references to the arXiv are in the form of URLs that are
spelled out but not hyperlinked.

The review makes many references to the arXiv, both explicit and implicit.
For example the paragraph,

    Since the paper was published, there have been a number of
    developments in the field. They include a few more geometric versions
    of Deligne's conjecture (as compared to the original Tamarkin version)
    by McClure and Smith, Kontsevich and Soibelman, and the reviewer,
    and further simplification (by Tamarkin and Tsygan) of Tamarkin's
    proof of the deformation quantization theorem, using Etingof-Kazhdan
    "dequantization" rather than quantization.

could be filled in with arXiv references as follows:

    Since the paper was published, there have been a number of
    developments in the field. They include a few more geometric versions
    of Deligne's conjecture (as compared to the original Tamarkin
    version [math.QA/9803025]) by McClure and Smith [math.QA/9910126],
    Kontsevich and Soibelman [math.QA/0001151], and the reviewer
    [math.QA/9807037], and further simplification [math.KT/0002116]
    (by Tamarkin and Tsygan) of Tamarkin's proof of the deformation
    quantization theorem [math.QA/9803025], using Etingof-Kazhdan
    "dequantization" [q-alg/9701038] rather than quantization.

Meanwhile Kontsevich explicitly mentions the arXiv in his paper:

    The story of this conjecture is quite dramatic. In 1994 E. Getzler and
    J.  Jones posted on the e-print server a preprint [hep-th/9403055] in
    which the proof of the Deligne conjecture was contained. Essentially
    at the same time M. Gerstenhaber and A. Voronov in [hep-th/9409063]
    made analogous claims.  The result was considered as well established
    and was actively used later.  But in the spring of 1998 D. Tamarkin
    observed that there was a serious flaw in both preprints. The cell
    decomposition used there turned out to be not compatible with the
    operad structure; the first example of wrong behavior appears for
    operations with 6 arguments.

At the end of the review, Voronov refers to formality theory as a
"rapidly developing field".   It is indeed developing faster than what the
traditional publication cycle can support.  This raises the question of
whether it is in the arXiv because it is developing quickly or vice-versa;
I'd like to think that the truth is some of both.  By contrast, the
idiosyncracies of Math Reviews are motivated by assuming a dichotomy
between preprints (meaning temporary, unrefereed drafts) and publications
(permanent, refereed final copies).  In my opinion this dichotomy has
become feeble.  I think that it is as important to preserve the math
arXiv as it is to archive any journal. At the same time, in areas such
as formality theory formal publication is little more than a formality.
(It didn't matter to the Fields Medal Committee that q-alg/9709040
was unpublished, and it didn't matter in Kontsevich's narrative that
hep-th/9409063 was published while hep-th/9403055 was not.)  This is a
shame in that formal publication is still one of the major venues for
peer review. So as I said in other postings, I think that peer review
needs a new formula.  And I think that informative reviews such as
Voronov's will eventually be part of that formula.

"This week in the mathematics arXiv" may be freely redistributed
with attribution and without modification.

[deletia --djr]
-- 
  /\  Greg Kuperberg (UC Davis)
 /  \
 \  / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/
  \/  * All the math that's fit to e-print *